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2018/4/14
*d¥*±h¥±kte & #X## EPEE sttt
'
EEE' .
slit -
4¥ HE ¥A a K¥+5
@ TakeTo 1024
Contents
0 .
t
'
7th # ksuc .
1 .
4¥ ' HtTHE a hat # at #¥x .
2 . Euler ¥ta a A¥# Aiz 'l¥s4t¥a .
3 .
PEEMEIE'EF# rmz .
4 ....
O .
t
'
7th # knuz
i 7th # sit
¥F⇒sb'ts3#¥h#
a # ¥n±kt9kh# Eat A "¥n2n3ta .
531 M =
s2 I a f¥tt9HlFh E = TM .
(Tangent bundle )
TM
~
# '
* Be...
"
Etants
"
n' 9th #
E¥4¥a '
EMIT E = Mx
Rr
it €53hMknt9Hv#n .
= th 2 As #± A t
'
9th # z Eta # At 9th
'
# said .
^3Ni'
.
4...
2) to Etna Itau tea IEIAA .
i ) El E = s
'
x IR tiztnktt
' '
.
3.
'
E. Ii
'
n' 7Hubt -
ha ( to ) { 53
'
'Ei7Hv±¥s to 2123 ...
95k¥ HE M a
"
#'s
"
a .
Zn F¥ Fk K Ls ERk2a3 .
4¥ K TM A
'
LFAA Azi 's b'
IF ,
M 2 'E#i3 It E¥#a9 .
931 .
s
'
a F¥ # a # a...
931 .
5 a t¥ # T5 is Etna itsu .
→ →
tl #aAts5 ,
5 Ii non -
zero then 7khE¥ b"
h¥k3 rtd .
→ Is
→
I→I ( b 'l
'
# 1£ zfnbt i...
Kd .
K - 52 T
'
a ¥¥ # It Etaa .
T2 = s
'
× s
'
,
TT2= Ts
'
×Ts
'
= -12×(112×112) F) ,
¥,
¥
% "
€a4=L⇐k .
574£ EIEHZFFHL 5...
( £4 e :X na £¥X )
-
n' 9th Eh it 3F¥eh# ± a n' 7th ±¥s * HYEXBFKEl
'
{ FR 's than ¥¥¥ .
- ni 9th # on # AA i tahr he ¥Eti...
1 .
4¥44 * A ah #¥ as tE±¥
"
HE # A f
"
Tsai
'
Fti tEF¥5 It
43$33 s ( II ( 2 k£ £ .
HE # East .
3. THE # M tansth # t : E ¥ M KKHZ .
Mn ITHEIII -
¥¥ a "¥tEi CCE ) e HIM ; @ ) { t.tk#2FhF5az
"
.
Az#± to A7Hv...
( "FEFE ) KERI - .
ITHEDI -
rmz
FYFE # M a Th Eat
'
-
¥¥ elf ,
Ma IKE K
£H3A#
Eta th 'm { 5-23 A' ¥2 at # FE .
BI .
Hals
'...
( "FEFE ) Thea I - .
IF # a I -
rmz
FYFE # a ] THE DI
'
-
¥¥ e it ,
M a TMNK Hk hT# z
t.tk#3FE7*tixti5Ai3ht*ExAS2FFx
.
HE ...
4¥ # ¥57 sit ( # HE )
Fy THE # M Ian "9th # E httlz .
Mn ITHEDI -
¥¥ a "¥tEi CCE ) e HEM ; @ ) { t.tk#2FhF5az
"
.
Az#± to ...
1) Euler #a
FYFE # Mta A±HH5hEi7Hv # t : E ¥ M lift .
E a Euler ¥57 It ,
ECE ) e
HTM
; 7L )
zi .
4nF {
IEAEFEN
.
D e ( ft ...
2)
Stiefel -
Whitney #A
FYFE # Mta i7Hv # t : E ¥ M KFH ,
E a Stiefel -
Whitney # A { Witt
)3F=
.
H .
WICE ) E H
:(
M ; 22...
3)
Chern #I
FYFE # Mta
kFE¥ai7Hv
# t : E #M kttl .
E a Chern ¥52 {
Cite
)
}F=°
H .
CICE
) E H
"
( M ; 2 ) ( i. 0,1 ,
...
,...
l 7¥¥uEgttkR¥ .
)
75k¥ # MKIH . Mta nsth # a
R#±¥k¥¥¥
{
tfk±t2B¥I
:
Vectt ) : Mtd →
Sets
a a
M i→ Vea (M ) =
{ E → M }/±
4...
2 .
Euler #E a t¥¥Ai z
'
pasta .
* .
) ] I It Bbl 55kt ¥EE¥ Hani Fta
.
Euler #aa#I¥Ai .
Rr
FFEHHSKK A 9th # TL :
E M K # L ,
Eo :=
E -
Mx { o }
SEE
,
Mta # Eva
"
a 7tN±'¥
"
s : M → E Ezy ,
 -
A...
the# 1 E bl
"
non -
zero section Etosit ,
ECE ) = 0 .
4¥ 's E A
"
Et a A Fsstt
'
ECE ) =
0 .
i
'
) s : M → Eo E non -
zero...
# I # 2 77 FE # Ma Euler #x { XCM) 5h32 .
←
Ma¥¥¥a
(
ECTM
) ,
[ M ] > =
XCM ) (
fuetlu
) =
XCM ) )
^Mn¥*
# a Euler #A
MnEl...
tz .
KTFIHF ( a # At ) if 8 a I # FK Bas't s 5-23 .
•
XC s
'
) =
4 -
6+4 =
2 .
•
•
•
He I # 2 gy ,
eCTs2 ) =
0 s if 1274¥ ...
( Fun { Hina Eatx )
-
7¥ 'HE ¥k ,
E. a ! EID
# In it to { ÷zGH=n izt '
teeth h3 . . . ( Ftxk Ethic ,zH2 ! )
-
cxl
A¥¥ Ai I...
HEE )
3. PEE '¥FE¥÷t¥ i= suz
per
¥ Et night # IT : E → M A
"
# AA b 'i5b '
# 2312 ;
non -
zero section s : M → Eo I zkzdlz # HE .
#
E
Rr i
:
-
Mx{o } :...
# EE As,
r=2 I M HE# FIFA'¥'t
Ikzeisza
.
sz ¥¥th#lz2£ 's .
( o )
Fai Ma Jake. As a # 95k M → Eo zhF3 .
;•,⇐#
EPi
, 1 1
'
-...
# baa,
#a5Eu it YBK Faa Eb ' ?
t#•
⇐
•
s
( )
¥¥#
" " " }
•
OK
of )
" "
a
FX
.
Fy.
• •
IFFFH, a TIEKF * Mx { o } e # Litt '...
# EE { it
'
Ma # IFAFI's little ,
EE'Re bi Mx{o } a As DE
FI # F3 A { EAF P¥f#£ c : cz ( µ ) →
2 2¥23 .
i.
⇐ . as
E
• • 2
...
# EE in F '¥#x c : GCM ) → 2 IF 2- cochair 2ta3 .
±5k
Sc(D3
) =
cc
D8
) =
0 ztay ,
2-
cocyde 2ta3 .
i.
€ . as
E
• • 2
T- •...
-
#k¥F# the Hit 5kt a 9th # a : E # M re HL ,
E n non -
zero section s : M → Eo It
M a v. 1
t.hn#a=tiaEeFhh4i=FFtathtiiI3
...
4 .
4¥ 't'¥¥B a
"
I #
"
Euler #A a Ih#
HE ## A -±hfH5hk n' 7th # t : E
*
M KJHL .
E a Euler ¥72
ECE ) e HRCM ; 2)
IF ,
E a non .
zero section E hF...
EL F¥k
r=dimM
at¥h¥ .
s : M → Eo e (E) =0
'
Ee !
.
:) Ma ✓ Manta it M { atafib 's . B
[ TI En # kktitvkt Xtistutsti 'E3 .
...
Stiefel .
Whitney #ZaIh#
a-
EL
FEEE ninth # E * M ⇐ HL ,
f , T
•
. - -
→
Ea k -
FFKYE VKCE ) E it E In k ] a
!
12k$ '&'IHA...
% WICE ) = 0 c⇒ E it A IHTHFIFAE
.
:) W , (E) =
0 # M
' ' '
± zi Vr .
it , (E) = VRCE ) a
section lot 2123 .
c ⇒ M
" '
s F...
Chern #2aIh*
4ft # KEEFE ninth # E
Is
M EHL ,
E a kBa¥,
k -
HER VKCE ) at ,
E In k ] a
1 -
12km , 'ItaA9Hua4E a 's Ai 3 73...
( £4 c ¥iF¥5 )
-
4¥ HE # a it n' 7th # a Btk¥E .
> Fy
' '
EFAAHEFZEEVH
'
'
{ tttttu ⇒ ,
Ali 's { FEI ,
-43 FED #x state
!
...
¥,¥¥e
Characteristic Classes
Milnor, Stasheff
微分位相幾何学
田村一郎
Lecture Notes in Algebraic Topology
Davis, Kirk
available online!
Thanked
@ Take To 9024 .
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Published on

2018/04/18 @ https://connpass.com/event/82142/

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特性類の気持ち

  1. 1. 2018/4/14 *d¥*±h¥±kte & #X## EPEE sttt ' EEE' . slit - 4¥ HE ¥A a K¥+5 @ TakeTo 1024
  2. 2. Contents 0 . t ' 7th # ksuc . 1 . 4¥ ' HtTHE a hat # at #¥x . 2 . Euler ¥ta a A¥# Aiz 'l¥s4t¥a . 3 . PEEMEIE'EF# rmz . 4 . 4¥ HE #a a ' ' IHI ' ' . * =a7¥Eirt Ih¥±± F) " ¥eF±t5 " 2 h#¥El£F . 5*h{ KIBE, tkukrt # Fta 1- t.EE#TEeIFFET In .
  3. 3. O . t ' 7th # knuz
  4. 4. i 7th # sit ¥F⇒sb'ts3#¥h# a # ¥n±kt9kh# Eat A "¥n2n3ta . 531 M = s2 I a f¥tt9HlFh E = TM . (Tangent bundle ) TM ~ # ' * Beta . I> y = ¥µTpM :# ' LIMIT p 1122 : 7 } hli ' - 4pm YY : EIIERIF ( pkaF¥isHvtEp¥l ) ← n7HH¥aaF#±⇒ ' ( ninth '¥ X e * an a × : t.Y.IT#ptas # HE . )
  5. 5. " Etants " n' 9th # E¥4¥a ' EMIT E = Mx Rr it €53hMknt9Hv#n . = th 2 As #± A t ' 9th # z Eta # At 9th ' # said . ^3Ni' . 4¥ , , A¥# Egg ;qµ*z . ' trivial Vector bundle ) By M = s ' In ELKE # . FZTHTEAKH F¥e*U×R' at 's . FT IT # RA ' # rtn9HlFkath¥kE£h3 . RR - R d s ' - • • • • • €1- - - 4) E = S ' x IR 2) E : open Mobius band C trivial ) ( non . Trivial )
  6. 6. 2) to Etna Itau tea IEIAA . i ) El E = s ' x IR tiztnktt ' ' . 3. ' E. Ii ' n' 7Hubt - ha ( to ) { 53 ' 'Ei7Hv±¥s to 2123 ttf lb 't FEE # a z # 2324M¥ the a 'k#¥ b 's HF " 0 { he ,2lF 's . qp¥. " rahman • n' 9th # A 3F÷kh# e i7HvBE¥n " tslikk TELEFE " .
  7. 7. 95k¥ HE M a " #'s " a . Zn F¥ Fk K Ls ERk2a3 . 4¥ K TM A ' LFAA Azi 's b' IF , M 2 'E#i3 It E¥#a9 . 931 . s ' a F¥ # a # a A . * • • ¥ " Fit 'U • ~ • = • • • • - - - TS ' 5×121 '
  8. 8. 931 . 5 a t¥ # T5 is Etna itsu . → → tl #aAts5 , 5 Ii non - zero then 7khE¥ b" h¥k3 rtd . → Is → I→I ( b 'l ' # 1£ zfnbt i Etsu ! ( taei 'EEAAF3 ) → € a 4- It fi - N KH AT ' 7ft (£EHBHA=) b " ZEZLF 5 . a # in 'a#⇒ -15¥ 5×1122
  9. 9. Kd . K - 52 T ' a ¥¥ # It Etaa . T2 = s ' × s ' , TT2= Ts ' ×Ts ' = -12×(112×112) F) , ¥, ¥ % " €a4=L⇐k . 574£ EIEHZFFHL 5123 . T -12=5×1122
  10. 10. ( £4 e :X na £¥X ) - n' 9th Eh it 3F¥eh# ± a n' 7th ±¥s * HYEXBFKEl ' { FR 's than ¥¥¥ . - ni 9th # on # AA i tahr he ¥Eti¥ K FR 23 . - A- a A 1 ¥ F Eta A I n 9th # 2 EB 't It " . A #¥ His Hv In a H hit " Al " he " { ' ¥ a 3 ft hi Makin . it 'E±¥ ) ¥ t Et a A Tt a 7th # it suzrt . EnKEEFE FE ( Bikaata ) t ÷s¥t# tkn .
  11. 11. 1 . 4¥44 * A ah #¥ as tE±¥ " HE # A f " Tsai ' Fti tEF¥5 It 43$33 s ( II ( 2 k£ £ .
  12. 12. HE # East . 3. THE # M tansth # t : E ¥ M KKHZ . Mn ITHEIII - ¥¥ a "¥tEi CCE ) e HIM ; @ ) { t.tk#2FhF5az " . Az#± to A7Hv # lift hit Ali h#Lz It 'Fi+t3ta . input output E | µ B " ) W ° DE ) e HIM ; %) n' ' 7th # . IKEA Ii Fs¥a¥t¥a .
  13. 13. ( "FEFE ) KERI - . ITHEDI - rmz FYFE # M a Th Eat ' - ¥¥ elf , Ma IKE K £H3A# Eta th 'm { 5-23 A' ¥2 at # FE . BI . Hals ' ) = | . ( k=° ) HKCT ) = | • ( k=o ) 0 ( k = 1 ) g ( k=l ) . ( k = 2) ; ( k= 2) , - - - - , • it. 52 -12
  14. 14. ( "FEFE ) Thea I - . IF # a I - rmz FYFE # a ] THE DI ' - ¥¥ e it , M a TMNK Hk hT# z t.tk#3FE7*tixti5Ai3ht*ExAS2FFx . HE ' ' HE a iktt Hit a Ifk k£3 bags a :c : / IK , < sd . this) = | . → 2 h=o ) ( ±*YYY*Yn⇒¥IFIt¥¥I*F¥*n#⇒ 0 ( k = 1 ) . → 2 ) ( k = 2) ] # EDI - #¥ k If Cup 45¥ K JR I'¥a tt3 Eh , t At2 K Then I - ¥4 zy t±# bits F¥a4⇐ z to .
  15. 15. 4¥ # ¥57 sit ( # HE ) Fy THE # M Ian "9th # E httlz . Mn ITHEDI - ¥¥ a "¥tEi CCE ) e HEM ; @ ) { t.tk#2FhF5az " . Az#± to A7Hv # lift hit At ht# z It 'Fi±t3€a . input output f Ka' tkn E ( µ "3 " ) W ° DE ) e HIM ; @ ) t.IE#iazY ( Helen ) i 7th # . ITHERI ' - ¥¥a¥t¥a .
  16. 16. 1) Euler #a FYFE # Mta A±HH5hEi7Hv # t : E ¥ M lift . E a Euler ¥57 It , ECE ) e HTM ; 7L ) zi . 4nF { IEAEFEN . D e ( ft E) = fte (E) I AYE 't ) 2 e ( EDF ) = ec E) . e (F ) 3 E : A- a A ⇒ EC E) = 0 4 E : E ATEGTE ⇒ ec E) = - ECE ) .
  17. 17. 2) Stiefel - Whitney #A FYFE # Mta i7Hv # t : E ¥ M KFH , E a Stiefel - Whitney # A { Witt )3F= . H . WICE ) E H :( M ; 22 ) ( i. 0,1 , ... ,r ) zi . 4nF { is '¥kF€n. 00 Wo( E) = 1 . 1 Wic# E) =ftw:( E) l⇐7*H± ) 2 W ( EDF ) =W(E) - WCF) ( w=1 + wit . - . + Was ) 3 W ,(8 , " ) = L ( Le HYRPIZ. ) =2z , 4=AiE )
  18. 18. 3) Chern #I FYFE # Mta kFE¥ai7Hv # t : E #M kttl . E a Chern ¥52 { Cite ) }F=° H . CICE ) E H " ( M ; 2 ) ( i. 0,1 , ... ,r ) zi . 4nF { IEAEFEN. @ Co (E) = 1 . D Cicft E) =ftc:(E) IEMIIHE) 2 c(EDF ) = CCE) . CCF) ( c=nta+ ... + cn ) 3 c.( 8 , ' ) = d ( de H4 # 2) a 2 , EIAIE )
  19. 19. l 7¥¥uEgttkR¥ . ) 75k¥ # MKIH . Mta nsth # a R#±¥k¥¥¥ { tfk±t2B¥I : Vectt ) : Mtd → Sets a a M i→ Vea (M ) = { E → M }/± 4¥ ' Kt ¥2 et, =aF¥t 't $5 IAEA 's " . P¥f4 na FFYEI '¥## a :{ : µf+E E a f* a Vectcm ) Vect CN ) f : M → N tr c c 3- A¥t#E¥n¥h÷k f* HTCMIA ) HTCN ;A ) 6 c # E) = ftctt ) " CCE ) ( ETYIIHI )
  20. 20. 2 . Euler #E a t¥¥Ai z ' pasta . * . ) ] I It Bbl 55kt ¥EE¥ Hani Fta .
  21. 21. Euler #aa#I¥Ai . Rr FFEHHSKK A 9th # TL : E M K # L , Eo := E - Mx { o } SEE , Mta # Eva " a 7tN±'¥ " s : M → E Ezy , - Annett -47*4 ( section ) 2n3 . EaThom*a : TC E) TIE ) A i* 9 HTE ,Eo ) > HRCE ) = Thom #¥ ' I s* HOCM ) - HTM )e a 1 ECE ) fruits . K -2,2 E a Euler #a el E) e HIM ;2 ) { 'E¥xF3 .
  22. 22. the# 1 E bl " non - zero section Etosit , ECE ) = 0 . 4¥ 's E A " Et a A Fsstt ' ECE ) = 0 . i ' ) s : M → Eo E non - zero section {(2 . e (E) a #¥aAikikZAn32 . ( E. Eo ) a cohomology ex , seq . LT , des HE ' ° )A A A A HTEEO ) HTE ) HMEO ) A s* s* HIM ) a ECE ) = 0 B'
  23. 23. # I # 2 77 FE # Ma Euler #x { XCM) 5h32 . ← Ma¥¥¥a ( ECTM ) , [ M ] > = XCM ) ( fuetlu ) = XCM ) ) ^Mn¥* # a Euler #A MnEler¥I7 EEEAA RB ) 97 F¥h# M a Euler #x XCM ) H , XCM ) = Io c- isi rank CHICM ) ) i 'a¥x±k3¥¥#k . 4¥ K M A " = # FIB '¥'t HE F¥ # At a ±n¥h± . (M ) = V - E + F . X aa¥nh##* ) Katana ) M#ah#¥D
  24. 24. tz . KTFIHF ( a # At ) if 8 a I # FK Bas't s 5-23 . • XC s ' ) = 4 - 6+4 = 2 . • • • He I # 2 gy , eCTs2 ) = 0 s if 1274¥ An . tkbtr theft 1 sy Ts2 If FIAA itsu ! - to , XCT ) = 0 g) , t*hL ECTT ' ) = 0 { ta2n3 . ( It ) iaiz $15 TTZ of EFAA iTa3 ' - st kFBt2u . ) ni 7th # a Et AAHI a TXFTEH't HE a- a A # at ,A¥h4 ) 2£23 AE'¥E A H¥skt= !
  25. 25. ( Fun { Hina Eatx ) - 7¥ 'HE ¥k , E. a ! EID # In it to { ÷zGH=n izt ' teeth h3 . . . ( Ftxk Ethic ,zH2 ! ) - cxl A¥¥ Ai I ÷*d ... ( SW #a . Chern#aaA¥# Air Er Hit ) he " k¥57 a chtttn { R > It 5k£ ?
  26. 26. HEE )
  27. 27. 3. PEE '¥FE¥÷t¥ i= suz
  28. 28. per ¥ Et night # IT : E → M A " # AA b 'i5b ' # 2312 ; non - zero section s : M → Eo I zkzdlz # HE . # E Rr i : - Mx{o } : 11h ftp.lntznjk -23 . it ? If : S µ EH #aAt£5€53h s It FTFSFZ ( A=aAita3t=aaXT¥FHt )
  29. 29. # EE As, r=2 I M HE# FIFA'¥'t Ikzeisza . sz ¥¥th#lz2£ 's . ( o ) Fai Ma Jake. As a # 95k M → Eo zhF3 . ;•,⇐# EPi , 1 1 ' - Mx{ 0 } "Ytiltii' ' l ' . ' ' ' ' , 1 . 1 i , , i t i , I I 1 itufAsI ML, ' , 1 1 1 1 1 1 • ' 1 *. :-/, =kE #a5¥u Its 's a # both M " '→Eo { 94-3 . t.sk the #a3⇐acz A±A5n. - .
  30. 30. # baa, #a5Eu it YBK Faa Eb ' ? t#• ⇐ • s ( ) ¥¥# " " " } • OK of ) " " a FX . Fy. • • IFFFH, a TIEKF * Mx { o } e # Litt 's e . ¥42 it IHNI't KFFAFE it An !
  31. 31. # EE { it ' Ma # IFAFI's little , EE'Re bi Mx{o } a As DE FI # F3 A { EAF P¥f#£ c : cz ( µ ) → 2 2¥23 . i. ⇐ . as E • • 2 T- • 1 ¥14×{0 } . ° • - 1 sf • : • PH. a• c = o 135 it " Ehn FEEEHFEAE ! at 0 not , Eitik s z FEBEAFZ PEETE ( obstruction ) 4 ' 'ta3 .
  32. 32. # EE in F '¥#x c : GCM ) → 2 IF 2- cochair 2ta3 . ±5k Sc(D3 ) = cc D8 ) = 0 ztay , 2- cocyde 2ta3 . i. € . as E • • 2 T- • 1 ;t.to:: • < • YT#ML• Letsie [ c ] EHYM ;2 ) an 's cohomology '¥kbtek3 . c it s NEK ' ) 'FkhFc3bt , [ c ] rshfustan ! ( EaAbI¥± )
  33. 33. - #k¥F# the Hit 5kt a 9th # a : E # M re HL , E n non - zero section s : M → Eo It M a v. 1 t.hn#a=tiaEeFhh4i=FFtathtiiI3 . stir #aFE £iF¥aAi it 3M£ a¥s 1 PEE '¥¥k [ c ] e HRCM ; 2) k£2 K£3 . > FT , Is : Mcr , → Eo [ c ] = 0 T c=o tss { a££FEBEaiE3 . A ' ht ' ' ) 'I > ! [ c ] =o 175 , - ¥4Ey£iE3izz ' ( r¥#a£ifEiEiE3 ! cost # au )
  34. 34. 4 . 4¥ 't'¥¥B a " I # "
  35. 35. Euler #A a Ih# HE ## A -±hfH5hk n' 7th # t : E * M KJHL . E a Euler ¥72 ECE ) e HRCM ; 2) IF , E a non . zero section E hF3Eh a * np¥a'¥¥A e - Exa3 ! CEIIAAIIE ) = KK £ , 2 Euler ¥52 { hE¥x at En . f3{ . ( E :#a A ⇒ ECE ) . . 0 ] if ha¥xb 's AA 5101473.
  36. 36. EL F¥k r=dimM at¥h¥ . s : M → Eo e (E) =0 ' Ee ! . :) Ma ✓ Manta it M { atafib 's . B [ TI En # kktitvkt Xtistutsti 'E3 . : ) rank ( TM ) = dim M F) , In # HA ' As ' I 'Is . XCS ' ) = 2 F) easy to Ita 's , 5 I KIF non - zero vector field ttzktsu . DK
  37. 37. Stiefel . Whitney #ZaIh# a- EL FEEE ninth # E * M ⇐ HL , f , T • . - - → Ea k - FFKYE VKCE ) E it E In k ] a ! 12k$ '&'IHA9kka¥E $ 's # 373mi ' . # . I i M p E a kikStiefel - Whitney ¥57, WRCE ) e HKCM ; 22 ) If , Vr. kt , (E) a MK 't a section ehF3Eh a * 1p¥IH÷**k ( mod 2) s - ZQFZ .
  38. 38. % WICE ) = 0 c⇒ E it A IHTHFIFAE . :) W , (E) = 0 # M ' ' ' ± zi Vr . it , (E) = VRCE ) a section lot 2123 . c ⇒ M " ' s F¥aAih3E±a sink - 7012€ , A⇐4eEs⇒. i. ¥hEI3 Ea ¥± # A ek3 . ← AEHAFEKXZHE's . ( ⇒ Mta KEE. ah . s°±i . ftp.g#$ygy,y*⇐&* ⇒ " ¥453 Ea¥± # signs . #
  39. 39. Chern #2aIh* 4ft # KEEFE ninth # E Is M EHL , E a kBa¥, k - HER VKCE ) at , E In k ] a 1 - 12km , 'ItaA9Hua4E a 's Ai 3 73mi ' - # . E a kihChan ¥57, CKCE) e HYM ; 2) It , Vr . kt , (E) a MMI a section ktthf * 1 EE '¥*k e - 2ixF3 .
  40. 40. ( £4 c ¥iF¥5 ) - 4¥ HE # a it n' 7th # a Btk¥E . > Fy ' ' EFAAHEFZEEVH ' ' { tttttu ⇒ , Ali 's { FEI , -43 FED #x state ! - ZEZE 3 Th Eat ' - EE Ka kER¥7 At A 31 , by " ) E #E4IF3 F¥f**k # took hEhia3 Fkttai , { it 4¥ 'tE#I ol ' ta5hn3 at 22€ AEK . - n' 7th 5ha Bnikb " It #sFaA a ] # Eaii - EF a 4h Task R3 aii FIFA tabspart at kit ¥-445 A 9th # c b ' Ft # Liliana €45613 . - # in 114+714 Blitz " 3 AIFAF a Iitt . " h543 faith " to "¥nERit3 . → 6^(18 ) : F¥PQzkE Grossmann 957.3kt# , A 7th Tan B¥A # Part
  41. 41. ¥,¥¥e Characteristic Classes Milnor, Stasheff 微分位相幾何学 田村一郎 Lecture Notes in Algebraic Topology Davis, Kirk available online!
  42. 42. Thanked @ Take To 9024 .

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